To determine the area of the remaining board after cutting out the largest possible circle from a square with a side length of 12 feet, let's follow these steps:
1. Calculate the area of the square:
The side length of the square is given as 12 feet. The area [tex]\( A_{\text{square}} \)[/tex] of a square is computed using the formula:
[tex]\[
A_{\text{square}} = \text{side length}^2
\][/tex]
Substituting the given side length:
[tex]\[
A_{\text{square}} = 12^2 = 144 \text{ square feet}
\][/tex]
2. Determine the radius of the largest possible circle:
The largest possible circle that can be cut from a square will have a diameter equal to the side length of the square. The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{\text{side length}}{2} = \frac{12}{2} = 6 \text{ feet}
\][/tex]
3. Calculate the area of the circle:
Using the formula for the area of a circle [tex]\( A = \pi r^2 \)[/tex] and given that [tex]\( \pi \approx 3.14 \)[/tex]:
[tex]\[
A_{\text{circle}} = 3.14 \cdot (6^2) = 3.14 \cdot 36 = 113.04 \text{ square feet}
\][/tex]
4. Calculate the area of the remaining board:
To find the area of the remaining board after the circle is cut out, subtract the area of the circle from the area of the square:
[tex]\[
\text{Remaining area} = A_{\text{square}} - A_{\text{circle}} = 144 \text{ square feet} - 113.04 \text{ square feet} = 30.96 \text{ square feet}
\][/tex]
Therefore, the approximate area of the remaining board is closest to [tex]\( 31 \text{ square feet} \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{31 \text{ square feet}} \][/tex]
